Vividness refers generically to any measure of a color's purity, colorfulness, or saturation.
There's more than one way to measure vividness, and these different methods result in color spaces with different qualities. Make It Colorful includes two measures: chroma and saturation.
Chroma is a fairly intuitive measure of vividness. Black, white and grays all have a chroma of 0, and all the pure colors on the hue spectrum have a chroma of 1. Everything else falls somewhere in between.
Although it is easy to understand, you'll notice in the color space slices above that chroma-based color space bounds are not simple squares, where any value for chroma and luminance are valid. Instead, there are some combinations of chroma and luminance which are invalid, in that if the color were converted to the RGB color space, at least one of the red, green, or blue channels would be outside the range [0, 1].
Saturation is not strictly a single measurement. Instead, each measure of luminance defines its own measure of saturation, which is based on the luminance at which the chroma value can be 1. Using this information, saturation essentially stretches chroma values out to fill the entire [0, 1] range. The result is that all combinations of saturation and luminance within the range [0, 1] are valid.